Mathematics Unit
A ratio is a comparison of the relationship between two quantities or categories of things. For example, a ratio might be used to compare the number of girls to the number of boys who are enrolled in a school.
There are three ways in which a ratio can be expressed. One point to keep in mind is that the numbers that make up the ratio should be expressed according to the way in which the information is stated in the original problem with the number that represents the first category being listed first and the number that represents the second category being listed second. Each of the three methods used for expressing ratios is demonstrated below to show the ratio of girls to boys who are enrolled at Richmond High School when the ratio is two girls for every three boys.
Method (1): One way to write a ratio is by using the word to in between the two numbers that represent the comparison being made. Thus, the ratio of girls to boys who attend
Richmond High School can be expressed as 2 to 3.
Method (2): A second way in which a ratio can be written is by placing a full colon (:) in between the two numbers that represent the comparison being made. Thus, the ratio of girls to boys who attend Richmond High School can be expressed as 2:3.
Method (3): The most common way to write a ratio is in fraction form. When writing ratios in fraction form, the number that represents the first category mentioned in the problem is written as the numerator and the number that represents the second category is written as the denominator. Thus, the ratio of girls to boys who attend Richmond High School can be expressed as
2
.
3
Note #1: When two numbers being used to indicate a ratio are written as a fraction, the fraction can be expressed in simplest form in order to make the problem easier to solve by using the greatest possible common factor to reduce the numerator and the denominator of the ratio numbers.
Example: There are 21 boys and 35 girls in a school choir. Therefore, the ratio of boys to girls is
21
. However, both 21 and 35 can be evenly divided by the common factor 7. As a result,
35 the original ratio expressed as a fraction can be reduced to lower terms:
21 ÷ 7 = 3
35 ÷ 7 = 5
Note #2: When the numerator of a ratio that is written in fraction form is larger than the denominator, it will form an improper fraction. When this happens, the fraction should be reduced to lowest terms, but left as an improper fraction such as
7
3
or
5
1
instead of converting
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Mathematics Unit it to a mixed numeral (
2
1
) or a whole number (5) as is normally done when expressing an
3 improper fraction in simplest form.
Example: There are 15 horses and 10 donkeys that live on a farm. Express the number of horses as compared to the number of donkeys in ratio form.
Step # 1: Place the number 15 in the numerator position and the number 10 in the denominator position to show the ratio as an improper fraction:
15
10
Step #2: Check to see if the ratio can be reduced to simpler form. Since both the numerator and the denominator that make-up the ratio can be evenly divided by the common factor 5, the original fraction can be reduced:
15 ÷ 5 = 3
10 ÷ 5 = 2
Step #3: Once the original improper fraction has been reduced to simplest form, leave it as an improper fraction:
3
2
When working problems that involve ratios, it is a good idea to label the numbers that make up the ratios in order to avoid making mistakes when solving a problem. The ratio of boys to girls at Richmond High School used earlier in this study-guide can be used as an example.
Example: There are 2 girls for every 3 boys who attend Richmond High School. Express the ratio of girls to boys in fraction form and label both numbers that make-up the ratio to indicate what they represent: girls 2
boys 3
Some ratio-proportion problems require that a ratio number be used that represents the combined total of the two categories being compared. The ratio number that represents the total can be created by simply adding the two ratio numbers that represent each of the ratio categories.
Example: If the ratio of girls to boys in a class is 2 to 3, what is the ratio number that represents the total number of students in the class?
ratio for girls 2
ratio for boys + 3
ratio for total students 5
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Mathematics Unit
A proportion is a comparison between two equal ratios. Two ratios are said to form an exact proportion if they represent the same fraction quantity. There are two methods that can be used in order to determine whether or not two ratios form an exact proportion.
Method #1: Reducing Each Ratio to Lowest Terms: Write both ratios in fraction form and reduce them to simplest form. If both ratios being compared can be reduced to the same fraction, the original ratios form an exact proportion.
Example: Compare the following two ratios:
16
and
8
20 10
Step #1: Analyze the first ratio written as a fraction to determine whether or not it is in simplest form. Since both 16 and 20 can be evenly divided by the common factor 4, the original ratio number
16
can be reduced:
20
16 ÷ 4 = 4
20 ÷ 4 = 5
Step #2: Analyze the second ratio written as a fraction to determine whether or not it is in simplest form. Since both 8 and 10 can be evenly divided by the common factor 2, the original ratio number
8
can be reduced:
10
8 ÷ 2 = 4
10 ÷ 2 = 5
Step #3: Once the ratios being compared have been reduced to simplest form, compare them to determine whether or not they are equal. In this example, both ratios can be reduced to
4
.
5
Therefore, the two original ratios form an exact proportion.
Method #2: Using the Cross-Multiplication Process: Write both ratios in fraction form and align them side-by-side. Cross-multiply the numerator of the first ratio times the denominator of the second ratio and compare it to the answer you get by cross-multiplying the denominator of the first ratio and the numerator of the second ratio. If the answers you get are the same, the original ratios form an exact proportion .
Step #1: Write the two ratios being compared in fraction form and place them side-by-side
16 8
20 10
Step #2: Cross-multiply the numerator of the first ratio (16) times the denominator of the second ratio (10) and the denominator of the first ratio (20) times the numerator of the second ratio (8).
16
20
8
10
16 x 10 = 160
20 x 8 = 160
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Mathematics Unit
Step # 3: Analyze the results of the cross-multiplication process you did in Step # 2. Since the product of both multiplication processes was 160, it proves that the two original ratios are in exact proportion to one another.
In translating a ratio-proportion word-problem, make sure that the categories that make-up each of the ratios being compared match exactly on both sides of the equal sign. An easy way to make sure you have set-up your ratios correctly is by labeling all of the numbers that make-up the ratios being compared.
Example: The ratio of the number of girls as compared with boys in a class is 2 to 3. The actual number of girls is 10 and the actual number of boys is 15. Write the two ratios as a proportion.
girls 2 = girls 10
boys 3 = boys 15
By using the cross-multiplication procedure discussed earlier in this study-guide, it is possible to determine the missing number in a ratio-proportion problem.
In performing such operations, a letter such as N can be used to substitute for the missing number you are trying to find. In the process of solving the problem, one ratio number will be multiplied times the unknown amount (N) and placed directly next to the letter N on one side of the equal sign (=) in an equation, and the product you get by multiplying the other two numbers will be placed on the opposite side of the equal sign. After the multiplication process has been completed, divide the number connected to the letter N into the number on the opposite side of the equal sign in order to find the missing number.
Example: If the ratio of horses to cows living on a farm is 5 to 3, how many horses are there if there are 24 cows?
Step #1: Write the ratio of horses to cows on one side of an equal sign and label the numbers that make-up the ratio: horses 5 =
cows 3 =
Step #2: Write the actual number of horses and cows as a ratio on the opposite side of the equal sign. Use the letter (N) to represent the missing quantity (the number of horses).
Remember to match the categories of the ratio numbers and the actual number of the things being compared in exactly the same way on each side of the equal sign. Use labels to be sure you are placing the numbers in their proper positions:
horses 5 = N horses
cows 3 = 24 cows
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Mathematics Unit
Step # 3: Check to see if the fraction that represents the ratio of horses to cows can be reduced.
Since there are no other common factors for 5 and 3 other than the number 1, the ratio cannot be reduced.
Step #4: Cross-multiply the numbers on either side of the equation (remember that one of the two numbers you will get by multiplying will end up next to the letter N):
horses 5 N horses
cows 3 24 cows
5 x 24 = 3 x N
120 = 3N
Step #4: Once the cross-multiplication process has been completed, divide the number connected to the letter N (3) into the number on the opposite side of the equal sign (120):
3
40
120
Note: The result of the final computation shows that if the ratio of horses to cows living on a farm is 5 to 3, and the actual number of cows is 24, then the actual number of horses is 40.
The same cross-multiplication and division process used when comparing two sets of ratios can be used to solve percent problems. In doing so, one of the ratios that is created to form an equation will involve some part of some total quantity that is identified in the problem being solved. The number representing the part is used in the numerator position of the ratio and the number representing the total quantity is used as the denominator on one side of the equal sign. The ratio on the opposite side of the equal sign will consist of the percent number in the problem and the number 100 (replaces the percent sign).
The percent number is used as the numerator of the ratio, and the number 100 is used as the denominator. When the problem is set-up, it will look like this:
Part Number = Percent Number
Total Quantity = 100
Note: There are a number of key words and symbols that can help identify what numbers belong in which positions in the two ratios being compared in percent problems:
1.The word of is often placed directly to the left side of the number that represents the total
quantity amount.
2.The word is or an equal sign (=) is often placed either directly to the right or left side of the
number that represents the part amount.
3. The word percent or a percent symbol (%) is placed directly to the right side of the number
that represents the percent amount.
4. The number 100 is always used to replace the term percent or the % symbol.
5. The letter N, or any other letter, can be used to replace the missing number in the percent problem being solved.
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Mathematics Unit
Example #1: ___________ is 50% of 80.
Step #1: Use the key words and symbols in order to place the numbers included in the problem in their correct positions in the ratio-proportion equation:
Part Number = Percent Number
Total Quantity = 100
1. The blank space represents the number you are trying to find and is located directly to the left of the word is. Therefore, the letter N should be placed in the Part Number position
2. The number 50 is placed directly to the right side of the percent symbol (%). As a result, the number 50 should be placed in the Percent Number position.
3. The word of is placed directly to the left side of the number 80. Thus, the number 80
should be placed in the Total Quantity position.
Note: After placing the numbers in their proper positions in the ratio-proportion formula, the equation should look like this:
N = 50
80 = 100
Step #2: Check to see if the ratio containing both numbers can be reduced to simpler form.
Since 50 is the greatest common factor for itself (50) and for 100, the ratio can be reduced:
50 ÷ 50 = 1
100 ÷ 50 = 2
Step #3: Use the cross-multiplication procedure discussed earlier in this study-guide in order to solve for the missing quantity (Part Number) represented by the letter N:
N 1
80 2
N x 2 = 80 x 1
2N = 80
Step #4: Divide the number 2 connected to the letter N into the number on the opposite side of the equal sign in order to solve the problem:
2
40
80
Note: The result of the final computation shows that 40 is 50% of 80.
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Mathematics Unit
When translating information from a word problem into the ratio-proportion formula, analyze the information by using the key words discussed earlier in this study-guide if possible.
Example #1: The distance Johnny can throw a baseball is 240 feet. This is 80% of the distance that his older brother can throw a baseball. How far can Johnny’s older brother throw a baseball?
Step #1: Use the key words and symbols in order to place the numbers included in the problem in their correct positions in the ratio-proportion equation: Part Number = Percent Number
Total Quantity = 100
1.
The number 240 feet is located directly next to the word is in the problem. Therefore, 240
is the part number.
2. The number 80 has a percent symbol next to it. As a result, 80 is the Percent Number.
3.
The word of refers to the total distance that Johnny’s older brother can throw a baseball,
which is the missing amount you are trying to find. Thus, the letter N represents the Total
Quantity amount in the equation.
4.
The number 100 replaces the percent symbol and should be placed directly under the
percent number in the equation.
Note: After placing the numbers in their proper positions in the ratio-proportion formula, the equation should look like this:
240 = 80
N = 100
Step #2: Reduce the ratio containing both the numerator and denominator to simplest form if possible. Since 20 is the greatest common factor for 80 and for 100, the ratio can be reduced:
80 ÷ 20 = 4
100 ÷ 20 = 5
Step #3: Use the cross-multiplication procedure discussed earlier in this study-guide in order to solve for the missing quantity (Percent Number) represented by the letter N:
240 4
N 5
N x 4 = 240 x 5
4N = 1200
Step #3: Divide the number 80 connected to the variable N into the number on the opposite side of the equal symbol (1200) in order to solve the problem:
4
300
1200
Note: The result of the final computation shows that 240 is 80% of 300.
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Mathematics Unit
Example #2: Emily weighs 120 pounds and Marta weighs 84 pounds. Emily’s weight is what percent of Marta’s weight?
Step #1: Using the terms is, of, and the = sign to identify the information in order to place numbers in their correct positions in the formula does not always work when solving word problems. Instead, it is sometimes necessary to analyze the information carefully using logic.
This is the case in example # 2 above:
1.
Emily’s weight (120 lbs.) is the total amount.
2.
Marta’s weight (80 lbs.) is being compared to Emily’s weight and is the part amount.
3.
The percent amount is what is being calculated. Therefore, the letter N is used in the
percent number position in the formula.
4.
As always, the number 100 is placed beneath the percent number in the problem.
After placing the numbers in their proper positions in the ratio-proportion formula, the equation should look like this:
84 = N
120 = 100
Step #2: Reduce the ratio containing both the numerator and denominator to simplest form if possible. Since 12 is a common factor for both 84 and 120, the ratio can be reduced:
84 ÷ 12 = 7
120 ÷ 12 = 10
Step #3: Position the two ratios on either side of an equal sign (=) and use the crossmultiplication procedure discussed earlier in this study-guide:
7 N
10 100
N x 10 = 7 x 100
10N = 700
Step #4: Divide the number 10 connected to the letter N into the number on the opposite side of the equal symbol (700) in order to solve the problem:
70
10
700
The ratio-proportion formula can be used to determine the length of the
side of one geometric figure that is missing when you know the length of the same side
(corresponding side) of another similar figure.
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Mathematics Unit
Geometric figures are considered to be similar when they have the same shape and when the corresponding angles formed where their sides meet measure the same number of degrees. It should be recognized that even though the angles measure the same number of degrees, the sides of similar geometric figures are different in length.
When solving problems in which you are trying to determine the length of the missing side of one of two similar geometric figures, you should form a ratio using two sides of one of the figures and compare it to the ratio of the same sides of the other figure. When doing this, the numbers representing the lengths of the sides that are similar should be placed in the same positions in both ratios. Use a letter such as N or X to represent the side of the geometric figure that is missing.
Example #1: The two triangles in this problem are similar. Determine the length of the side of the triangle that is missing represented by the letter N.
20
N
32
8
Step #1: Form a ratio using the two sides that are identified in the first triangle: 20
32
Step #2: Reduce the ratio if possible in order to make the problem easier to solve. Since 4 is a common factor for both 20 and 32, the original ratio number can be reduced:
20 ÷ 4 = 5
32 ÷ 4 = 8
Step #3: Form a ratio using the two sides of the second figure that correspond with the sides of the first figure by placing the number and letter representing the sides of the second figure in the
same positions as they are in the first ratio. The side that is missing is represented by the letter N:
Step #4: Position the two ratios on either side of an equal sign (=): 5 = N
8 = 8
Step #5: Use the cross multiplication process discussed earlier in this study-guide:
5 N
8 8
8 x N = 5 x 8
8N = 40
Step #6: Divide the number 8 connected to the letter N into the number on the opposite side of the equal sign (40):
5
8 40
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Mathematics Unit
Note: The length of the side that is missing is 5. You might have noticed that once the original ratio
20/32 was reduced to simplest form, it had the same denominator (8) as the ratio to which it was being compared. Therefore, the numerators of both ratios ended up being the same as well (5).
Example #2: Determine the length of the side that is missing from the largest of the two similar rectangles shown to the right:
17 p
12 30
Step #1: Form a ratio using the two sides that are identified in the first rectangle: 12
17
Step #2: Reduce the ratio if possible in order to make the problem easier to solve. Since there are no common factors for 12 and 17 other than the number 1, the ratio cannot be reduced and therefore is already in simplest form.
Step #3: Form a ratio using the lengths of the two sides of the larger rectangle by placing the sides that correspond with the smaller triangle in the same positions in the second ratio. The side of the larger triangle that is missing is represented by the letter P: 30
P
Step #4: Position the two ratios on either side of an equal sign: 12 = 30
17 = P
12 30
17 P
17 x 30 = 12 x P
510 = 12P
Step #6: Divide the number 12 connected to the letter P into the number on the opposite side of the equal sign (510).
42 .
5
12 510
Note: The result of the computation shows that the length of the missing side of the larger of the two similar rectangles being compared is 42.5.
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